Tag - Hall algebras

Naihuan Jing: Characters of GLn(q) and vertex operators

3rd March, 2024

Irreducible characters of the finite group GLn(q) were determined by Green in a remarkable paper that has influenced representation theory greatly. In this talk, I will discuss a vertex algebraic approach to construct and compute all complex irreducible characters of GLn(q). Green's theory is recovered and enhanced under the realization of the Grothendieck ring of representations R(G)=⨁​n≥0​​R(GLn(q)) as two isomorphic Fock spaces. Under this picture, the irreducible characters are realized by the Bernstein vertex operators for Schur functions, the characteristic functions of the conjugacy classes are realized by the vertex operators for the Hall-Littlewood functions, and the character table is completely given by matrix coefficients of vertex operators of these two types. This offers a simplification to identify the Fock space R(G) as the Hall algebra of symmetric functions. We will also discuss how to compute the characters in general.

Alistair Savage: Affine Hecke algebras and the elliptic Hall algebra

13th May, 2021

The elliptic Hall algebra has appeared in many different contexts in representation theory and geometry under different names. We will explain how this algebra is categorified by the quantum Heisenberg category, which is a diagrammatic category modelled on affine Hecke algebras. This categorification can be used to construct large families of representations for the elliptic Hall algebra.

Shiquan Ruan: Hall polynomials for tame type

18th August, 2016

In this talk we will show that Hall polynomial exists for each triple of decomposition sequences which parameterize isomorphism classes of coherent sheaves of a domestic weighted projective line X over finite fields. These polynomials are then used to define the generic Ringel–Hall algebra of X as well as its Drinfeld double. Combining this construction with a result of Cramer, we show that Hall polynomials exist for tame quivers, which not only refines a result of Hubery, but also confirms a conjecture of Berenstein and Greenstein.